When one talks about a “self-evident truth,” what exactly is one talking about?
In one sense, it is “self-evidently true” that when one looks at an object — say, this pint glass next to me as I type — I see that it is a pint-glass. It is “self-evidently true” that I am looking at a pint-glass (putting to aside worries of Cartesian demonic deception), because I do not perform an inference. My perception of the pint-glass is not the conclusion of an argument, based on premises. It is a paradigm case of non-inferential knowledge.
But in another sense, this perceptual knowledge is not “self-evident,” if by that we mean knowledge that does not depend on any further presuppositions. For the contrary is the case: a great deal of background knowledge must be presupposed in order for me to see the pint-glass — for example, I must have the concept of “pint-glass” and know how to apply that concept. Even the transparent cases of analytic propositions (“a vixen is a fox”, “the sum of the interior angles of a Euclidean triangle is 180 degrees”, “every effect has a cause”) presuppose as their respective background an adequate grasp of the concepts involved.
It is sometimes said that if a proposition is self-evidently true, then nothing can be done which would show it to be to true to someone who denied it. But this is not quite right. What is right is that a proposition is self-evidently true, then it cannot be demonstrated from some other premises nor arrived at through generalizations — it is not grounded in either deduction or induction, one might say.
But that does not mean that one cannot resort to all sorts of other arguments or thought-experiments that disclose that the proposition is self-evidently true. A classic example of this is Descartes’s famous “I think, therefore I am.” This is not the result of inference or observation, yet Descartes spends a great deal of time setting the stage to prepare the reader for this truth and to see it as self-evident. For this reason, “I can’t convince of you of this, because it’s self-evidently true” should not be accepted without criticism.
Because of this distinction between non-inferential knowledge and presuppositionless knowledge, accepting the importance of the former does nothing to settle whether or not we ought to be committed to the latter. The failure to see this is what Sellars called “the Myth of the Given,” which is the original sin of rationalism and empiricism alike.